Immersed Spheres of Finite Total Curvature into Manifolds

acceptedyear: 2011 journal: Advances in Calc. Var. abstract: We prove that a sequence of, possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact
riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse
to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$
and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature.
We prove moreover that if the sequence stays within a class $\gamma$ of $\pi_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.